Question

are the following true or false? explain

a) If a sequence (an) converges, then the series ∑an also converges

b) if a series ∑an converges, then the sequence (an)converges to zero.

c) if a sequence (an) converges to zero, then the series ∑ an converges.

Answer 1

for a)
imagine \(\{a_n\}\) as \(n\to\infty\) converges to 1
then as an infinite series we are adding\[1+1+1+...\]as the final terms
would that sum converge, or diverge?

Answer 2

for c) try convergence of 1/n

Answer 3

@ turningtest, it'll diverge because it goes up to infinity?

Answer 4

correct :)

Answer 5

Nevermind.

Answer 6

I think "Near" is correct. Don't know how to explain that very well though.

Answer 7

yes you were correct @bmp

Answer 8

Haha. Yeah, I will think better about it. But I think that if the n-th term of a series don't go to 0, as series is a summation, as n grows larger, the sum will never converge. So, at some point the element \( a_{n} \) has to go to 0, so, the sequence will converge.

Answer 9

sort of the counter-point to my argument about the first one diverging :)

Answer 10

Yup, so I think that it would be F T F, as both of you (@TuringTest and @experimentX) hinted on the other questions.

Answer 11

for b, I think if by adding up infinite number, answer will approach the limit. I'm assuming test for divergence?

Answer 12

not sure what you mean by that @roastedchickenwings